A circle has centre (3,0) and radius 5. The line y = 2x + k intersects the circle in two points. Find the set
of possible values of k, giving your answers in surd form.

A circle with center and radius has equation

If we substitute in this equation, we have

This equation has two solutions (i.e. the line intersects the circle in two points) if and only if the determinant is greater than zero:

The expression simplifies to

The solutions to the associated equation are

So, the parabola is negative between the two solutions:

x=6 and x=3

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